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A new diabatic molecular representation in He+-he charge transfer collision
Votumc 85, number 2
A NEW DIABATIC MOLECULAR
8 J.muxy
CHCMICAL PHYSICS LETTERS
REPRESENTATION
IN He+-He
CHARGE TRANSFER
1981
COLLISION
Rccewed 31 July 1981
A “frozen~,rbl~aI”dhb~tlc bass 113sbeen constructed to study rhc I&z+-t!c rcsona~t charge trzmslcr colhslon in .I closc* \‘-rl rotar~onal couplJngs are reasonably free from the mcorrect asymplolrc coupled rmpact paramcrcr treatment Th~hr bshavlour (= R-‘) Compsrlson wnh expcrrmcnr at large scatccrrng angles demonstnrcs swuficrnt unprovcnlcnt over earher work
I. Introduction This work forms part of a program bcmg pursued r.n our group for sometune [J -31 for search of a surtablc diabntrc molecular representatron for use m atomrc cohrsrons The specrfic problem chosen 1s tJre resonant charge transfer m coJbs~ons of He+ eons with He, which was first shown by Jxhten 141 to be quote appropnately descnbable by a drnbarrc molecular picture. It IS well known that due to a large number of (avoided) curve crossmgs the apphCatJOJl of the close-coupled perturbed-stationary-states (PSS) method to this problem 1s enormously drfficult. It was shown rn our last work [3] that a molecular orbrtrd representatron consrstmg of “frozen” single Slater determmants formed with a mmJmaJ basrs set formed a good appro.umahon to the radral diabatic representahon defied by Smith IS]. In the present work rt 1s demonstrated that a very simple modificatron, namely the use of spur-coupled functrons instead of smgle determJnantaJ functions, results rn a remarkable rmprovement over the earher work. We were mostly mterested in the effect of t.hrs modlfied basis on the rotatronril coupbng between
the C and JJ states, and hence the calcularrons were done only at 3 keV laboratory ton energy where for close colhsrons the effect of rotntronal couphngs has long been known to be srgnlficant [6,7].
2. Theory
\ve are usmgatomlc umts throughout except where othcrwrse stated If one det’mes a cIasss~caJtraJectory R = R(r) for the nucIcar motton, the ttme-dependent Schriidmger equatron for the electromc matron IS
rw1 I al+, R(r))liU= ff,,\P(r. R(t))
0)
We ROWexpand the total wavcfunctron cated bssls set
W, R(O)=
* m a trun-
6 c,(r) OJr, R),
where the $1 are electromc basrs functrons III a molccuIe-fLued reference frame. Substttutton of (2) m (I) leads to the usual set of coupled equatrons IV
* Permanent address: Department of Ccneral Physics and XRays, Indian AssocJation for the CultJvatlon of Science. Calcutta 700032. India.
1dcL/dr
k = I, 2,. , ,v. (3) 219
In the impact parameter appro?.unatron seal nuclear traJectory ISgven by
Table 1 Blolecular orbaxIs ~1 used for constwang funcuons 0, as dcscnied IIItbc text
the clas-
R=b+ut,
(4)
a vta -=- +- vb -_a at R aR R2 a0
(5)
eq (3) becomes
XI
1 2
% 5
:$
} 3‘1,= 1 84. Siso = 1 68
43
r% g x’u
;i:
} tl,=
3dag
$, s = 1 92. szpo = 0 52
x
4ia,
r,s=
5
I dc/dt = (H + P + Q)c,
(6)
where c IS a column vector matrices defmed as
6 I
B 1”
8
r
9
“u
to
ng
and H, P and Q are square
HAI = Ml H,,W,
px, = (u~~~)w-~
Q,, =
(7)
Choice of dtabattc bass and solutto~tof the coupled equatrons. In a modrfication over our earlier work [3] we construct spur-coupled funcrlons
= il~~lo~ 0, =
Exponents R 2 20
optumzed
1 92.r2,=z
at
14
192,t2po=048
6
a/am,
(~1/R~)w-la/aol~>.
0, = {lU&lC7”X,)
tbc spmcouplcd
Symmetry Of@)
I
where b IS the unpsct parameter and u the velocity of relative motion of the nuclei. Thus alaf = u-OR, puttmg Z = ur and assuming azunuthal symmetry
Thus
8 January 1982
CHEhlICAL PHYSICS LETTERS
Volume 85. number Z
here the diabatic of the form
IS”, ‘n,]‘r,,
<,I + ifog $ouf
&/dt
= (H + Q)c.
(IO)
1 and 2 bemg degenerate at large R with the He+(ls) + He(lsl), we have the boundary condrhon States
mcident channel
basis IJI usmg
IC,(-==)I
%l”
c,(--)=O,
(u symmetry),
{la,(lo,~,j lx”, ‘ll,}“Z,,‘n,
I
= IC,(--)I
= 2-1’2,
k=3,.
,lO
The coupled equatrons were solved subject to the above boundary condrtron and the unttanty con&-
(g symmetry)
tion (8)
As m ref. 131, the molecular orbrtals lap, lu, and yI were constructed usrng linear combmatlons of mrmmal Slater-type functions Table I hsts the orbltals together with the exponents vanattonally optrmrzed and “frozen” at R 2 30. The dlabatlc basis
F
vector was now defmed as
E31-
W
RI = U WHk
RI,
Ic&)l’
= 1
(all t).
which was conserved to ~thm <, 3 parts in 1000 in calculation Details of computational procedures and numerical methods can be found in earlier papers
(99)
where 9 and + are column vectors, S(R) is the
3. Results and discussions
Schnudt orthonormabzatron matrrx and U = U(R 2 20) IS the “frozen” umtary matrix dragonaliztng S(R)+at large R. By surular arguments as m ref [3] rt IS seen that thrs basrs can be regarded as a close appro.xunatron to the standard radral drabatrc representation of Smrth [5], so that the radral coupling terms m eq (6) vamsh, reducting it to the form
The basis set was constructed and the potential and rotational coupling matrices set up in a Burroughs 6700 computer using 48-bit single precisron arithmetic, and they were used as input to solve the coupled equations in an IBM 370/168 computer using 64-bit double prection.
230
\\/I
Volume 85. number 2
zoo-
100 -
50 -
%7
‘\
020n t I’ IO-
OS-
02-
0 II
CHEMICAL PHYSICS LEITERS
\
\
‘\
2= H,,
*-._
--
----__-_-_--L---
H.3
HI,
I
10
u-
20
I
30
R
40
50
60
F@ 1 D~go~~Ipotenual matrLx elements of the u sutes bnes, ru states, broken hnes. n, states
rull
8 January 1982
Fogs 1 and 2 show the &agonal potential matrix elements of the u and the g states rcspectwely. USC of the spmtoupled functions to construct the drabatic basis is unmedlately seen to result III the following consequences. (I) For the -Cu and C, states 1 and 2 we obtamcd a separated-atom energy of 4.8380 au which is much closer to the Hartrce-Fock energy than IS the value gven by Evans and Lane [IO] and m our earher work [3] _We thmk this to be due to use of openshell functions (II) The crossmg between the two lowest Ig states which appeared to occur at R = 1.4 m our last work [3] as well as m an earher work of Olson [ 1I] was tifted to R = 0 5. Due to the pecuhar behavior of the states +e contauung the 4fu, orbltal (see ref. [I21 tn tlus context) some new tntertuedlate-range crossmgs were obtatned. A few typlwl off-dragonal potential matrt.. elements are shown m fig. 3, theu general behavior 1s found to be sumlar as m ref. 131. (m) Angular momentum couplmgs were now present between UN the Z-n pans. Fig. 4 shows the elestates. It wdl be seen that ments between the X -n except for the (4-JOf co~pling. the other elements dcvlate httle from the correct asymptottc (-R-‘) behavior. It may not be possible to draw any general conclusions from thus, but It is expected that the present basis would give a better account of the ro-
LA 00
Fy 3. Offdugoml Fig 2 Same as m fg 1, for the g states
43
50
potenwl
100 R
I50
200
matrw elements 1125.Hz6.
Ha6,
and H68
231
CHEMICAL
Volume 85 number 2
PHYSICS
‘0°1 70-
0 c) f z 0 T P
16101
I
30-
& ZO-
10 00
1 100
50
150
200
R l-16
4
Rot~t~oml
= (2 lo)
(4.10).
coup& elements (R’/vb)Qkl, (6.10) and (8.10)
with (b.0
tattonal couphng effects upon the coLhs~on. In fig. 5 we show the locatrons of the extrema III the chargetransfer probabdrty curves obtamed by Nagy et al. [6] agamst the present theoretrcal results together wrth the results obtamed earher III ref [3] and by McCarroll and Pmcentuu [7] _Towards large vahres of
8 January 1982
E0, where the “dephasmg” is qydicant, the present results are undoubtedly much supenor to the rwo other cahxlahons Our results mdrcate that no irnportant non-adrabatrc couphngs have been left out, and hence the claim made in ref. [7] of the tmportance of II-A couphngs at these ener@es is not borne out u-r thrs work. In conclusion, we note that the molecular basis used m thrs work is a good approxrmatron to the standard radraf dlabatrc representation of Snuth [S] , and at least tn tlus case of He+-He scattenng the rotatronal coupling terms are reasonably free from the mcorrect asymptotic behavror (- R-t) observed m many diababc basrs calculatrons (cf ref. [ 131). If thrs latter property IS found to be saustied for other systems as well, thrs m Itself would make drabarrc basrs expansrons of thrs type a strong candidate for more extensive use as a molecular wavefunctron representatron m atormc colhstons
Acknowledgement Part of the work has been supported by a grant from Indran Space Research Orgamzatron, Department of Space, Government of Indra. The authors are mdebted to Professor H S. Taylor of the University of Southern Cabforma, Los Angeles, for grant of computer tune under NSF Grant -‘CHE-7910583 for the rest of the work
,
0-
LETTERS
I I I I
o-
References
[ 11 S S Bhnttxharya
o-
!Ot
O4
I
5
I
b INDEX
I
I
7 8 OF EXTREMA
I
9
Fig 5 LocatIons of the exuema of the charge-transfer probabIry at Ion energy 3 keV mdeved from left to right. Cycles are the data porn& of Nsey et nl [6] IXI-bne c-e. present results, dot-dashed cume. rcf (31. broken carve, ref [7j
232
and T E Rm Dasudar. J Phys 88 (1975) 1522 [21 T R R~I Dastldx and S S Bhattnchzuyn. Indun J Phys 50 (1976) 731 [ 31 T K R~I Dastrdar. R Rw Dastxiar and hl Bose, Chem Phys 43 (1979) 183. [4] W Llchten.Phks Rev 131 (1963) 229 [S] F T Smlth.Phys Rev 179 (1969) 111 [Ci] S W Nqy, S.hl Fernandes and E Pollxk. Phys Rev A3 (1971) 280 [7j R McCarroU and R D Pucenruu, J. Phys B4 (1971) 1026 [8] E E Ndutm. III Cbemlsche Elcmentarprozesse, cd H Hartmann (Sponger. BerIm, 1968) [9] L Wdets and S J. Wallace. Phys Rev 169 (1968) 84 [lo] S A Evans and N F. Lane. Phys Rev A8 (1973) 1385 [ 111 R E Olson, Phys Rev A5 (1972) 2094 [ 121 JP Cnuyacq. m EkCtiOmC and atomx colhslons. ed G Water (North-Holland, Amsterdam, 1978) 1131 J B Delos.Comman At hlol. Phys 10 (1981) 211
A NEW DIABATIC MOLECULAR
8 J.muxy
CHCMICAL PHYSICS LETTERS
REPRESENTATION
IN He+-He
CHARGE TRANSFER
1981
COLLISION
Rccewed 31 July 1981
A “frozen~,rbl~aI”dhb~tlc bass 113sbeen constructed to study rhc I&z+-t!c rcsona~t charge trzmslcr colhslon in .I closc* \‘-rl rotar~onal couplJngs are reasonably free from the mcorrect asymplolrc coupled rmpact paramcrcr treatment Th~hr bshavlour (= R-‘) Compsrlson wnh expcrrmcnr at large scatccrrng angles demonstnrcs swuficrnt unprovcnlcnt over earher work
I. Introduction This work forms part of a program bcmg pursued r.n our group for sometune [J -31 for search of a surtablc diabntrc molecular representatron for use m atomrc cohrsrons The specrfic problem chosen 1s tJre resonant charge transfer m coJbs~ons of He+ eons with He, which was first shown by Jxhten 141 to be quote appropnately descnbable by a drnbarrc molecular picture. It IS well known that due to a large number of (avoided) curve crossmgs the apphCatJOJl of the close-coupled perturbed-stationary-states (PSS) method to this problem 1s enormously drfficult. It was shown rn our last work [3] that a molecular orbrtrd representatron consrstmg of “frozen” single Slater determmants formed with a mmJmaJ basrs set formed a good appro.umahon to the radral diabatic representahon defied by Smith IS]. In the present work rt 1s demonstrated that a very simple modificatron, namely the use of spur-coupled functrons instead of smgle determJnantaJ functions, results rn a remarkable rmprovement over the earher work. We were mostly mterested in the effect of t.hrs modlfied basis on the rotatronril coupbng between
the C and JJ states, and hence the calcularrons were done only at 3 keV laboratory ton energy where for close colhsrons the effect of rotntronal couphngs has long been known to be srgnlficant [6,7].
2. Theory
\ve are usmgatomlc umts throughout except where othcrwrse stated If one det’mes a cIasss~caJtraJectory R = R(r) for the nucIcar motton, the ttme-dependent Schriidmger equatron for the electromc matron IS
rw1 I al+, R(r))liU= ff,,\P(r. R(t))
0)
We ROWexpand the total wavcfunctron cated bssls set
W, R(O)=
* m a trun-
6 c,(r) OJr, R),
where the $1 are electromc basrs functrons III a molccuIe-fLued reference frame. Substttutton of (2) m (I) leads to the usual set of coupled equatrons IV
* Permanent address: Department of Ccneral Physics and XRays, Indian AssocJation for the CultJvatlon of Science. Calcutta 700032. India.
1dcL/dr
k = I, 2,. , ,v. (3) 219
In the impact parameter appro?.unatron seal nuclear traJectory ISgven by
Table 1 Blolecular orbaxIs ~1 used for constwang funcuons 0, as dcscnied IIItbc text
the clas-
R=b+ut,
(4)
a vta -=- +- vb -_a at R aR R2 a0
(5)
eq (3) becomes
XI
1 2
% 5
:$
} 3‘1,= 1 84. Siso = 1 68
43
r% g x’u
;i:
} tl,=
3dag
$, s = 1 92. szpo = 0 52
x
4ia,
r,s=
5
I dc/dt = (H + P + Q)c,
(6)
where c IS a column vector matrices defmed as
6 I
B 1”
8
r
9
“u
to
ng
and H, P and Q are square
HAI = Ml H,,W,
px, = (u~~~)w-~
Q,, =
(7)
Choice of dtabattc bass and solutto~tof the coupled equatrons. In a modrfication over our earlier work [3] we construct spur-coupled funcrlons
= il~~lo~ 0, =
Exponents R 2 20
optumzed
1 92.r2,=z
at
14
192,t2po=048
6
a/am,
(~1/R~)w-la/aol~>.
0, = {lU&lC7”X,)
tbc spmcouplcd
Symmetry Of@)
I
where b IS the unpsct parameter and u the velocity of relative motion of the nuclei. Thus alaf = u-OR, puttmg Z = ur and assuming azunuthal symmetry
Thus
8 January 1982
CHEhlICAL PHYSICS LETTERS
Volume 85. number Z
here the diabatic of the form
IS”, ‘n,]‘r,,
<,I + ifog $ouf
&/dt
= (H + Q)c.
(IO)
1 and 2 bemg degenerate at large R with the He+(ls) + He(lsl), we have the boundary condrhon States
mcident channel
basis IJI usmg
IC,(-==)I
%l”
c,(--)=O,
(u symmetry),
{la,(lo,~,j lx”, ‘ll,}“Z,,‘n,
I
= IC,(--)I
= 2-1’2,
k=3,.
,lO
The coupled equatrons were solved subject to the above boundary condrtron and the unttanty con&-
(g symmetry)
tion (8)
As m ref. 131, the molecular orbrtals lap, lu, and yI were constructed usrng linear combmatlons of mrmmal Slater-type functions Table I hsts the orbltals together with the exponents vanattonally optrmrzed and “frozen” at R 2 30. The dlabatlc basis
F
vector was now defmed as
E31-
W
RI = U WHk
RI,
Ic&)l’
= 1
(all t).
which was conserved to ~thm <, 3 parts in 1000 in calculation Details of computational procedures and numerical methods can be found in earlier papers
(99)
where 9 and + are column vectors, S(R) is the
3. Results and discussions
Schnudt orthonormabzatron matrrx and U = U(R 2 20) IS the “frozen” umtary matrix dragonaliztng S(R)+at large R. By surular arguments as m ref [3] rt IS seen that thrs basrs can be regarded as a close appro.xunatron to the standard radral drabatrc representation of Smrth [5], so that the radral coupling terms m eq (6) vamsh, reducting it to the form
The basis set was constructed and the potential and rotational coupling matrices set up in a Burroughs 6700 computer using 48-bit single precisron arithmetic, and they were used as input to solve the coupled equations in an IBM 370/168 computer using 64-bit double prection.
230
\\/I
Volume 85. number 2
zoo-
100 -
50 -
%7
‘\
020n t I’ IO-
OS-
02-
0 II
CHEMICAL PHYSICS LEITERS
\
\
‘\
2= H,,
*-._
--
----__-_-_--L---
H.3
HI,
I
10
u-
20
I
30
R
40
50
60
F@ 1 D~go~~Ipotenual matrLx elements of the u sutes bnes, ru states, broken hnes. n, states
rull
8 January 1982
Fogs 1 and 2 show the &agonal potential matrix elements of the u and the g states rcspectwely. USC of the spmtoupled functions to construct the drabatic basis is unmedlately seen to result III the following consequences. (I) For the -Cu and C, states 1 and 2 we obtamcd a separated-atom energy of 4.8380 au which is much closer to the Hartrce-Fock energy than IS the value gven by Evans and Lane [IO] and m our earher work [3] _We thmk this to be due to use of openshell functions (II) The crossmg between the two lowest Ig states which appeared to occur at R = 1.4 m our last work [3] as well as m an earher work of Olson [ 1I] was tifted to R = 0 5. Due to the pecuhar behavior of the states +e contauung the 4fu, orbltal (see ref. [I21 tn tlus context) some new tntertuedlate-range crossmgs were obtatned. A few typlwl off-dragonal potential matrt.. elements are shown m fig. 3, theu general behavior 1s found to be sumlar as m ref. 131. (m) Angular momentum couplmgs were now present between UN the Z-n pans. Fig. 4 shows the elestates. It wdl be seen that ments between the X -n except for the (4-JOf co~pling. the other elements dcvlate httle from the correct asymptottc (-R-‘) behavior. It may not be possible to draw any general conclusions from thus, but It is expected that the present basis would give a better account of the ro-
LA 00
Fy 3. Offdugoml Fig 2 Same as m fg 1, for the g states
43
50
potenwl
100 R
I50
200
matrw elements 1125.Hz6.
Ha6,
and H68
231
CHEMICAL
Volume 85 number 2
PHYSICS
‘0°1 70-
0 c) f z 0 T P
16101
I
30-
& ZO-
10 00
1 100
50
150
200
R l-16
4
Rot~t~oml
= (2 lo)
(4.10).
coup& elements (R’/vb)Qkl, (6.10) and (8.10)
with (b.0
tattonal couphng effects upon the coLhs~on. In fig. 5 we show the locatrons of the extrema III the chargetransfer probabdrty curves obtamed by Nagy et al. [6] agamst the present theoretrcal results together wrth the results obtamed earher III ref [3] and by McCarroll and Pmcentuu [7] _Towards large vahres of
8 January 1982
E0, where the “dephasmg” is qydicant, the present results are undoubtedly much supenor to the rwo other cahxlahons Our results mdrcate that no irnportant non-adrabatrc couphngs have been left out, and hence the claim made in ref. [7] of the tmportance of II-A couphngs at these ener@es is not borne out u-r thrs work. In conclusion, we note that the molecular basis used m thrs work is a good approxrmatron to the standard radraf dlabatrc representation of Snuth [S] , and at least tn tlus case of He+-He scattenng the rotatronal coupling terms are reasonably free from the mcorrect asymptotic behavror (- R-t) observed m many diababc basrs calculatrons (cf ref. [ 131). If thrs latter property IS found to be saustied for other systems as well, thrs m Itself would make drabarrc basrs expansrons of thrs type a strong candidate for more extensive use as a molecular wavefunctron representatron m atormc colhstons
Acknowledgement Part of the work has been supported by a grant from Indran Space Research Orgamzatron, Department of Space, Government of Indra. The authors are mdebted to Professor H S. Taylor of the University of Southern Cabforma, Los Angeles, for grant of computer tune under NSF Grant -‘CHE-7910583 for the rest of the work
,
0-
LETTERS
I I I I
o-
References
[ 11 S S Bhnttxharya
o-
!Ot
O4
I
5
I
b INDEX
I
I
7 8 OF EXTREMA
I
9
Fig 5 LocatIons of the exuema of the charge-transfer probabIry at Ion energy 3 keV mdeved from left to right. Cycles are the data porn& of Nsey et nl [6] IXI-bne c-e. present results, dot-dashed cume. rcf (31. broken carve, ref [7j
232
and T E Rm Dasudar. J Phys 88 (1975) 1522 [21 T R R~I Dastldx and S S Bhattnchzuyn. Indun J Phys 50 (1976) 731 [ 31 T K R~I Dastrdar. R Rw Dastxiar and hl Bose, Chem Phys 43 (1979) 183. [4] W Llchten.Phks Rev 131 (1963) 229 [S] F T Smlth.Phys Rev 179 (1969) 111 [Ci] S W Nqy, S.hl Fernandes and E Pollxk. Phys Rev A3 (1971) 280 [7j R McCarroU and R D Pucenruu, J. Phys B4 (1971) 1026 [8] E E Ndutm. III Cbemlsche Elcmentarprozesse, cd H Hartmann (Sponger. BerIm, 1968) [9] L Wdets and S J. Wallace. Phys Rev 169 (1968) 84 [lo] S A Evans and N F. Lane. Phys Rev A8 (1973) 1385 [ 111 R E Olson, Phys Rev A5 (1972) 2094 [ 121 JP Cnuyacq. m EkCtiOmC and atomx colhslons. ed G Water (North-Holland, Amsterdam, 1978) 1131 J B Delos.Comman At hlol. Phys 10 (1981) 211